Inverse Laplace transform

Inverse Laplace transform

Mellin's inverse formula

An integral formula for the inverse Laplace transform, called the Bromwich integral, the Fourier–Mellin integral, and Mellin's inverse formula, is given by the line integral:

$\mathcal{L}^{-1} \{F(s)\} = f(t) = \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)\,ds,$

where the integration is done along the vertical line Re(s) = γ in the complex plane such that γ is greater than the real part of all singularities of F(s). This ensures that the contour path is in the region of convergence. If all singularities are in the left half-plane, or F(s) is a smooth function on - ∞ < Re(s) < ∞ (i.e. no singularities), then γ can be set to zero and the above inverse integral formula above becomes identical to the inverse Fourier transform.

In practice, computing the complex integral can be done by using the Cauchy residue theorem.

It is named after Hjalmar Mellin, Joseph Fourier and Thomas John I'Anson Bromwich.

Post's inversion formula

An alternative formula for the inverse Laplace transform is given by Post's inversion formula.

• Inverse Fourier transform

References

• Davies, B. J. (2002), Integral transforms and their applications (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95314-4
• Manzhirov, A. V.; Polyanin, Andrei D. (1998), Handbook of integral equations, London: CRC Press, ISBN 978-0-8493-2876-3

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